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Non-linear objectives in mechanism design Shuchi Chawla University of Wisconsin – Madison FOCS 2012

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So far today… Revenue & Social Welfare This talk: Non-linear functions of type & allocation Question: how well can we optimize in strategic settings? Do Bayesian assumptions help? Shuchi Chawla: Non-linear objectives

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Algorithmic mechanism design Three desiderata: Computational efficiency Incentive compatibility Optimize/approximate objective Main theme in AMD: all three not always achievable together What should we give up? Shuchi Chawla: Non-linear objectives

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AMD tradeoffs Shuchi Chawla: Non-linear objectives Black-box

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AMD tradeoffs Shuchi Chawla: Non-linear objectives Black-box Single-parameter : each agent has a single value Monotone objectives : unilateral increase in an agents value causes OPT to allocate more to the agent IC condition : unilateral increase in an agents value results in larger allocation Prior-free Bayesian (sometimes)

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Rest of this talk Part I The makespan objective Impossibility of black-box reductions for makespan Part II Bayesian truthful approximations for makespan Other non-linear objectives; Open problems Shuchi Chawla: Non-linear objectives

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Part I.1: Minimizing makespan Shuchi Chawla: Non-linear objectives

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Scheduling to minimize makespan n jobs, m machines Jobs have different runtimes on different machines Makespan = completion time of last job Shuchi Chawla: Non-linear objectives J1J2J3J4 M1 M2 M3 Makespan Unrelated instance

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Scheduling to minimize makespan Strategic setting [Nisan Ronen99]: Machines are selfish workers; jobs runtimes are private Mechanism = (schedule, payments to machines) Machines objective: maximize payment – work done Want assignment+payments to induce truthtelling Shuchi Chawla: Non-linear objectives

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Why makespan? Important CS problem Captures the difficulty with non-linear objectives A single agent can disproportionately affect objective Has received the most attention in AGT Shuchi Chawla: Non-linear objectives

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J1J2J3J4 M1 M2 M3 Single-parameter makespan Each machine has a speed; each job has a size Runtime of job j on machine i = (size of j)/(speed of i) Monotone objective Shuchi Chawla: Non-linear objectives Makespan Related instance

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A history of prior-free scheduling Truthful approximations for related machines Archer-Tardos01 : constant approx Dhangwatnotai et al.08 : PTAS Unrelated machines: upper & lower bounds Nisan-Ronen99 : m approximation Nisan-Ronen99 : lower bound of 2 Christodoulou et al.07 : 2.41 ; Koutsoupias-Vidali07 : 2.61 Mualem-Shapira07 : randomized, fractional mechanisms Ashlagi-Dobzinski-Lavi09 : lower bound of m for anonymous mechanisms Shuchi Chawla: Non-linear objectives

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Bayesian model for scheduling Unrelated setting: Running time of every job on every machine drawn independently from known distribution Related setting: Speed of every machine drawn independently from known distribution; jobs sizes fixed Objective: Expected min makespan Shuchi Chawla: Non-linear objectives

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Part I.2: Black-box transformations Shuchi Chawla: Non-linear objectives

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Black-box transformations Shuchi Chawla: Non-linear objectives Input v Allocation x Payment p GOAL: for every algorithm, transformation preserves quality of solution and satisfies incentive compatibility. (cf. Nicoles talk)

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Black-box transformations Social welfare: can transform any approx. algorithm into BIC mechanism with no loss in expected performance. [Hartline-Lucier10, Hartline-Kleinberg-Malekian11, Bei-Huang11] Is this possible for other objectives? Makespan: For any polytime BIC transformation, there is a makespan problem and algorithm such that mech.s expected makespan is polynomially larger than alg.s. [C.-Immorlica-Lucier12] Shuchi Chawla: Non-linear objectives NO!

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Single-parameter makespan x1x1 m machines, machine i has speed v i ~ F i n jobs, size of job j is x j x 2 x3x3 x4x4 collection F of feasible assignments Shuchi Chawla: Non-linear objectives

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Proof outline Define makespan instance (feasibility constraints, value distribution). Find algorithm with low expected makespan. Use monotonicity condition to show any BIC transformation has high expected makespan. Key issue: Transformation must rely on algorithm to understand/satisfy feasibility constraint That is, transformation must return an allocation that it observes the algorithm return Shuchi Chawla: Non-linear objectives

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Makespan Instance feasibility set F = {at most one job per machine} m machines, speeds v i ~ Uniform{low = 1, high = α} m/2 jobs, small size x j = 1 m 1/2 jobs, large size x j = α Shuchi Chawla: Non-linear objectives

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Approximation Algorithm 1.If (m/2 ± m 3/4 ) machines report high speed, assign large jobs to fast machines (at random) assign small jobs to slow machines (at random) assign NO job to all remaining machines 2.Else assign all jobs randomly Shuchi Chawla: Non-linear objectives

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Approximation Algorithm high speedslow speeds Note 1: By Chernoff, expected makespan is low. Note 2: Expected allocation is not monotone. Shuchi Chawla: Non-linear objectives

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Transformation To fix non-monotonicity, must more often: 1.allocate nothing to low speed machines, or 2.allocate something to high speed machines. Shuchi Chawla: Non-linear objectives

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Transformation Query v: pretend some low machines are high and vice versa... Input v: Each fast machine gets large job with probability m -1/2 then with high probability, makespan is high. Shuchi Chawla: Non-linear objectives

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Transformation Query v: pretend number of high machines deviates from expectation.. Input v: Each machine gets large job with probability m -1/2 then with high probability, makespan is high. Shuchi Chawla: Non-linear objectives

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Recap and other results For any BIC transformation, there is an alg. such that the transformations makespan is polynomially larger than the algorithms even when the algorithm is a constant approximation What about other non-linear functions? Ironing doesnt work Gap increases with non-linearity Shuchi Chawla: Non-linear objectives [C.-Immorlica-Lucier12]

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Non-linear objectives in mechanism design Shuchi Chawla University of Wisconsin – Madison Part II

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Recap of part I A representative non-linear objective: makespan Black-box transformations are essentially impossible for makespan: objective function increases by polynomial factor Shuchi Chawla: Non-linear objectives Black-box

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Part II.1: Bayesian approximation for makespan Shuchi Chawla: Non-linear objectives

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Recall: scheduling to minimize makespan n jobs, m machines Jobs runtimes drawn from known indep. distributions Makespan = completion time of last job Prior-free setting: any anonymous truthful mechanism is at best an m approximation. Shuchi Chawla: Non-linear objectives J1J2J3J4 M1 M2 M3 Makespan

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A truthful mechanism: MinWork For every job: Assign the job to the machine that reports the lowest runtime Pay the machine the jobs running time on its second best machine m Second-price payments: induce truthtelling Makespan sum of best runtimes of all jobs total work done in optimal schedule m x optimal makespan m-approximation to makespan Shuchi Chawla: Non-linear objectives

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Overcoming the lower bound Ashlagi et al.s lower bound of m for makespan Ordered instance: machine i is better than machine i+1 for all jobs Running times within 1+eps of each other Any truthful mechanism must allocate all jobs to machine 1 How do Bayesian assumptions help? Knowledge of distribution => we can penalize allocations that are always bad for the given instance A priori identical machines: bad instances have extremely low probability Shuchi Chawla: Bayesian scheduling 31

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Prior-independent approximation Unknown Bayesian prior, but belongs to some nice family In particular, the runtime of a job j is identically distributed on every machine. That is, machines are a priori identical However, any instantiation of runtimes is an unrelated instance Result: There exists a truthful prior-independent mechanism that achieves an O(n/m) approximation to expected makespan (*) [C.-Hartline-Malec-Sivan12] Shuchi Chawla: Non-linear objectives (cf. Tims talk)

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Benchmark Hindsight OPT For any instantiation, finds the optimal makespan OPT 1/2 Discards m/2 machines randomly For any instantiation, finds optimal makespan over remaining machines For many distributions, OPT 1/2 ~ constant. OPT Key property: min over 2 draws ~ 2 times a single draw Includes all MHR distributions, e.g. uniform, exponential, normal,… Shuchi Chawla: Non-linear objectives

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How to design a truthful multi-parameter mechanism? A simple powerful class: affine maximizers Maximize an appropriate linear a.k.a. affine function Essentially, an extension of VCG For example: Can assign costs to some outcomes, and, minimize total (work – cost) Can forbid certain outcomes by setting cost = Can assign more weight to the work of some agents than that of others Shuchi Chawla: Non-linear objectives

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The MinWork mechanism again Essentially VCG: schedule every job on its best machine Observe: job js runtime in MinWork job js runtime in OPT Furthermore, every job goes to a random machine If jobs were to be distributed uniformly across machines, we would get good makespan However, balls-in-bins analysis some machine has O(log m/log log m) jobs Shuchi Chawla: Non-linear objectives

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The MinWork (k) mechanism Find a min-size matching between jobs and machines that assigns at most k jobs to each machine. Claim: MinWork(k) is truthful Proof: It is VCG over a restricted domain. Shuchi Chawla: Non-linear objectives

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The MinWork (k) mechanism Find a min-size matching between jobs and machines that assigns at most k jobs to each machine. Claim: MinWork(k) is truthful Claim: MinWork(10) gets a constant approximation Obs1: The schedule is almost balanced Obs2: Every job still goes to roughly its best machine Shuchi Chawla: Non-linear objectives

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Obs2: the last entry procedure Fix job j and imagine adding it last in a greedy fashion. Shuchi Chawla: Non-linear objectives MinWork(3) schedule for all but job j MinWork(3) schedule sorted by js preferences Machine full Space available, so j goes here

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Obs2: The last entry procedure Fix job j and imagine adding it last in a greedy fashion. The probability that j goes to one of its top i machines is at least 1-(1/k) i MinWork(k) places j in an even better position Key claim: Placing j on its i th best machine is no worse than placing 5 i independent copies of j on their best (of n/2) machines Shuchi Chawla: Non-linear objectives

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MinWork (10) analysis Job js runtime in MinWork(k) max 5^i independent copies js runtime in OPT 1/2 5 i times js runtime in OPT 1/2 Here i is an exponential random variable; Note: E[5 i ] = constant. MinWork(10)s makespan 10 E[max j (js runtime in MW)] constant times OPT 1/2 Shuchi Chawla: Non-linear objectives Stochastic dominanc e

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Key technical claim Placing j on its i th best machine is no worse than placing 5 i independent copies of j on their best (of n/2) machines Shuchi Chawla: Non-linear objectives Expt. 1 n copies of js runtime Expt. 2 5 i /2 blocks n/2 copies of js runtime i th min over n copies max over 5 i mins over n/2 copies

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Recap and other results Machines a priori identical, few jobs O(1) prior-independent approximation: MinWork(k) O(1) OPT 1/2 Compare to Bulow-Klemperers result for revenue with k items: VCG O(1) OPT less k agents Jobs are also a priori identical: multi-stage mechanisms Prior-ind. O(log m) approximation to OPT 1/2 Prior-ind. O((log log m) 2 ) approx to OPT for MHR distributions Shuchi Chawla: Non-linear objectives [C.-Hartline-Malec-Sivan12] Hindsight-OPT 1/2 (needs regularity)

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Part II.2: Other objectives & open problems Shuchi Chawla: Non-linear objectives

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Open problems for makespan O(1) prior-ind. approximation for non-identical jobs Bayesian approximation for non-identical machines Will need to use the knowledge of prior Even logarithmic approx is non-trivial A potential approach: charge a prior-dependent amount for placing each additional job on a machine Approximation for small-support priors LP based? Shuchi Chawla: Non-linear objectives

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Other non-linear objectives Max-min fairness in scheduling a.k.a. load balancing Prior-free PTAS for related setting [Epstein-van Stee10] Unrelated approximation? Max-min fairness in welfare a.k.a. the Santa Claus problem Not monotone! Single-parameter Bayesian approx? Shuchi Chawla: Non-linear objectives Min makespan

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Conclusions Non-linear objectives in general much harder than social welfare Mild stochastic assumptions can help us circumvent strong impossibility results Multi-parameter mechanisms are difficult to understand, but affine maximizers is a powerful subclass. Lots of nice open problems! Shuchi Chawla: Non-linear objectives

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